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J. Llibre, Xiang Zhang (2010)
Rational first integrals in the Darboux theory of integrability in C^nBulletin Des Sciences Mathematiques, 134
AJ Miranda, EC Rizzioli, MJ Saia (2013)
Stable singularities of co-rank one quasi homogeneous map germs from $$(C^{n+1}; 0)$$ ( C n + 1 ; 0 ) to $$(C^n; 0)$$ ( C n ; 0 ) , $$n = 2; 3$$ n = 2 ; 3JP J. Geom. Topol., 13
J. Llibre, Marcelo Messias, A. Reinol (2015)
Normal Forms for Polynomial Differential Systems in ℝ3 Having an Invariant Quadric and a Darboux InvariantInt. J. Bifurc. Chaos, 25
G. Vallis (1988)
Conceptual models of El Niño and the Southern OscillationJournal of Geophysical Research, 93
F. Dumortier, Joan Ferragud, J. Llibre (2006)
Qualitative Theory of Planar Differential Systems
C. Gibson, K. Wirthmüller, A. Plessis, E. Looijenga (1976)
Topological Stability of Smooth Mappings
J. Llibre, Marcelo Messias, P. Silva (2008)
On the global dynamics of the Rabinovich systemJournal of Physics A: Mathematical and Theoretical, 41
R. Thom (1969)
Ensembles et morphismes stratifiésBulletin of the American Mathematical Society, 75
J. Llibre, Xiang Zhang (2012)
On the Darboux Integrability of Polynomial Differential SystemsQualitative Theory of Dynamical Systems, 11
J. Llibre, Marcelo Messias (2009)
Global dynamics of the Rikitake systemPhysica D: Nonlinear Phenomena, 238
J. Guckenheimer, P. Holmes (1983)
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 42
J. Llibre, Marcelo Messias, P. Silva (2011)
Global dynamics of stationary solutions of the extended Fisher–Kolmogorov equationJournal of Mathematical Physics, 52
A. Krishchenko, K. Starkov (2008)
Localization of Compact Invariant Sets of Nonlinear Time-Varying SystemsInt. J. Bifurc. Chaos, 18
E. Lorenz (1963)
Deterministic nonperiodic flowJournal of the Atmospheric Sciences, 20
J Llibre, M Messias, AC Reinol (2015)
Normal forms for polynomial differential systems in $$\mathbb{R}^3$$ R 3 having an invariant quadric and a Darboux invariantInt. J. Bifurcation Chaos, 25
J Llibre, X Zhang (2009)
Darboux theory of integrability in $$\mathbb{C}^n$$ C n taking into account the multiplicityJ. Diff. Eqn., 246
N. Grulha (2011)
STABILITY OF THE EULER OBSTRUCTION OF A FUNCTION, 17
J. Llibre, Marcelo Messias, P. Silva (2012)
Global Dynamics in the Poincaré ball of the Chen System having Invariant Algebraic SurfacesInt. J. Bifurc. Chaos, 22
A. Kanatnikov, A. Krishchenko (2009)
Localization of invariant compact sets of nonautonomous systemsDifferential Equations, 45
J. Llibre (2004)
Chapter 5 Integrability of polynomial differential systems, 1
H. Whitney (1965)
Tangents to an Analytic VarietyAnnals of Mathematics, 81
J. Llibre, Xiang Zhang (2009)
Darboux theory of integrability in Cn taking into account the multiplicityJournal of Differential Equations, 246
J. Llibre, Marcelo Messias, P. Silva (2010)
Global Dynamics of the Lorenz System with Invariant Algebraic SurfacesInt. J. Bifurc. Chaos, 20
J. Llibre, Regilene Oliveira (2015)
Quadratic systems with invariant straight lines of total multiplicity two having Darboux invariantsCommunications in Contemporary Mathematics, 17
S. Strogatz (2016)
Nonlinear Dynamics and Chaos with Student Solutions Manual
J. Jouanolou (1979)
Equations de Pfaff algébriques
J. Llibre, Marcelo Messias, A. Reinol (2014)
Darboux invariants for planar polynomial differential systems having an invariant conicZeitschrift für angewandte Mathematik und Physik, 65
S. Wiggins (1989)
Introduction to Applied Nonlinear Dynamical Systems and Chaos
Marcelo Messias, A. Reinol (2016)
Integrability and Dynamics of Quadratic Three-Dimensional Differential Systems Having an Invariant ParaboloidInt. J. Bifurc. Chaos, 26
Stephen Wiggins (1988)
Global Bifurcations and Chaos
R. Euzébio, J. Llibre (2014)
Periodic solutions of el nino model through the vallis differential systemDiscrete and Continuous Dynamical Systems, 34
J. Llibre, C. Valls (2011)
Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systemsZeitschrift für angewandte Mathematik und Physik, 62
J. Llibre, R. Ramírez, N. Sadovskaia (2014)
Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian MechanicsJournal of Dynamics and Differential Equations, 26
J. Brasselet, L. Tráng, J. Seade (2000)
Euler obstruction and indices of vector fieldsTopology, 39
J. Nuno-Ballesteros, B. Oréfice, J. Tomazella (2013)
The Bruce-Roberts number of a function on a weighted homogeneous hypersurfaceQuarterly Journal of Mathematics, 64
J. Brasselet (2009)
Vector fields on Singular Varieties
J. Milnor, P. Orlik (1970)
Isolated singularities defined by weighted homogeneous polynomialsTopology, 9
In this paper we give the normal form of all polynomial differential systems in $$\mathbb {R}^3$$ R 3 having a weighted homogeneous surface $$f=0$$ f = 0 as an invariant algebraic surface and characterize among these systems those having a Darboux invariant constructed uniquely using this invariant surface. Using the obtained results we give some examples of stratified vector fields, when $$f=0$$ f = 0 is a singular surface. We also apply the obtained results to study the Vallis system, which is related to the so-called El Niño atmospheric phenomenon, when it has a cone as an invariant algebraic surface, performing a dynamical analysis of the flow of this system restricted to the invariant cone and providing a stratification for this singular surface.
Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals
Published: Jun 27, 2017
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